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Patterns & Relations/Data & Probability Front Matter Content Page Contents 1 2 Math Place Components for the Patterns & Relations/ Data & Probability Kit 4 Patterns & Relations/Data & Probability Overview



Contents 2 Math Place Components for the Patterns & Relations/ Data & Probability Kit 4 Patterns & Relations/Data & Probability Overview 10 Getting Started with Patterns and Relations 11 Inviting Patterns and Relations into the Classroom 15 Introducing Patterns and Relations 87 Introducing Equality and Inequality 132 Getting Started with Data and Probability 133 Inviting Data and Probability into the Classroom 135 Introducing Data 185 Introducing Probability 209 References

Math Place Components for the Patterns & Relations/Data & Probability Kit Read Aloud Texts Four Read Aloud texts are included to set a whole-class focus for learning and to provide realistic contexts for the math and help students to connect with it. Big Book The Patterns, Relations, Data, and Probability big book (with an accompanying digital version and 8 little book copies) is used to develop spatial reasoning and to create context for the math. Math Little Books Two math little books (8 copies of each) are used in guided math lessons with small groups for focused and differentiated instruction tailored to the needs of the students. They also offer opportunities to observe and assess students as they apply math concepts in problem-solving situations. 2 Patterns & Relations/Data & Probability

Teacher’s Guide A Teacher’s Guide supports teachers in building students’ conceptual understanding of math by providing hands-on learning experiences, using a variety of concrete materials and tools. This allows students to apply all of the curricular competencies as they solve problems. • Lessons include an About the Math section, which incorporates recent research to explain math concepts and why they are so critical to students’ current and future learning. • Detailed three-part lesson plans include rich problems and many opportunities for collaborative learning, communication of ideas, independent problem solving, and practice. The consolidating prompts and discussions are designed to connect students’ mathematical thinking and bring clarity to key mathematical concepts. • The three-part lessons offer suggestions on how to differentiate the learning to meet the specific needs of all students. • Activities develop mental math strategies based on conceptual understanding. Many visualization activities are included to support and develop students’ spatial reasoning skills. • Lessons support assessment for learning by offering suggestions on how to assess through observations, conversations, and products. There are also ‘Teacher Look-Fors’ to further support assessment and evaluation, and to serve as a guide for co-constructing success criteria with your students. • Further Practice and Reinforcement activities offer students the opportunity to practise newly acquired skills. • Math Talks provide support for posing comments and questions that promote interactive talk. • Blackline Masters (BLMs), such as creating patterns and find the missing number, are included in the Reproducibles guide and can easily be used to prepare for lessons. There are also some graphic organizers which help students record and organize their observations and mathematical thinking. In addition, all BLMs are available digitally on the Teacher’s Website. Teacher’s Website A variety of online resources, including Digital Slides, are available to support instruction and students’ problem solving. Also included are a digital version of the big book and modifiable Home Connections letters and Observational Assessment Tracking Sheets. Overview Guide A digital Overview Guide provides support for teaching all areas of math in Math Place, Grade One. The guide offers background information including the role of problem solving, visualization, and spatial reasoning in mathematics, and ideas for developing habits of mind, growth mindsets, and positive attitudes towards math. It includes assessment strategies and ways to differentiate to meet the needs of all students in your classroom. In addition, the Overview Guide outlines and explains the various high-impact instructional approaches used in the resource. 3

Patterns & Relations/Data & Probability Overview What Are Patterns and Relations? Patterns are sequences that repeat, grow, or shrink according to certain rules. These rules represent the ‘relations’ or relationships between the elements in the pattern. Algebra is about relationships, including the relationships found in various patterns. Beatty & Bruce explain that “patterning activities are introduced in elementary school so students can think about relationships between quantities early in their math education, which is intended to help them transition to formal algebra in middle school and high school” (Beatty & Bruce, 2012, p. 1). Students learn about algebraic relationships more formally in grade four. It is their experiences with patterns and relations in the primary grades that help to prepare them. Grade one students identify, describe, create, translate, and extend a variety of patterns, and also describe the pattern rules. They explore different repeating representations of patterns (e.g., concrete, pictorial, numerical) and make connections among them. Experiences with number patterns also extend students’ understanding of number relationships as they recognize patterns inherent to our base ten system. As students develop their understanding of numbers through studying patterns, they recognize the concepts of equality and inequality as they determine whether addition and subtraction expressions are equivalent or not, and use appropriate symbols to represent the concepts (=, ≠). They also investigate change in quantity by concretely and verbally describing how one quantity can change into another, either by adding or subtracting more to a given quantity. What Are Data and Probability? The learning standards in Data and Probability offer many opportunities to explore real-world problems and make connections to students’ everyday lives since people are continually exposed to data through advertising, news, polls, and social media. Students need to recognize that data can be collected in different ways and is often used to answer questions and make decisions in our everyday lives. By learning to represent data and interpret the results, students recognize the relationships among different sets of data. This knowledge can help support their thinking in different curriculum areas (e.g., Science), as well as help them develop critical-thinking skills. According to Marian Small, probability “is the study of measure of likelihood for various events or situations” (Small, 2009, p. 544). Young students think about familiar events and the likelihood that they will occur. This requires learning the accompanying vocabulary so students can describe likelihood and make comparisons. In grade one, students learn words such as ‘never,’ ‘sometimes,’ ‘always,’ ‘more likely,’ and ‘less likely’ to describe the probability of familiar events 4 Patterns & Relations/Data & Probability

occurring. Exploring probability allows students to make predictions and develop critical-thinking skills that will support their thinking in different curriculum areas (e.g., Science, English Language Arts). Integrating Number Throughout the Strands Number is the foundation for all mathematical understanding and permeates all curriculum strands. Helping students make connections among concepts they explore in Patterns and Relations and Data, and in other strands, will reinforce their mathematical understanding and support flexibility in their thinking. For example, skip-counting and operational sense relate directly to patterns and relations as students identify and perform the operations required to extend a number pattern or determine equality. Small notes that, “if we can connect a new idea being taught to related ideas that have been previously learned, it is more likely that the new knowledge will be assimilated” (Small, 2013, p. 18). Spatial Reasoning Spatial reasoning involves “the locations of objects, their shapes, their relations to each other, the paths they take as they move” (Newcombe, 2010, p. 30). It plays an integral role in all mathematical learning. For example, students can develop a strong understanding of equality and inequality if they can see the equivalence represented with concrete materials. This helps students develop the mental images that support their abilities to visualize mathematical concepts in a meaningful way. The Patterns and Relations and Data strands offer teachers a variety of ways to help students develop spatial reasoning skills. Patterning requires students to use visual-spatial skills as well as number sense. The graphical representations that students investigate in data are highly visual and allow students to make comparisons and draw conclusions. The Importance of Multiple Representations Communicating and representing are curricular competencies that focus on having students make their thinking visible. Using multiple representations allows students to make connections among concepts and offers differentiation for students who may use different approaches to solving problems. Small notes that “the more flexible students are in recognizing alternative ways to represent mathematical ideas, the more likely they are to be successful in mathematics” (Small, 2013, pp. 24–25). Providing opportunities for students to represent their thinking in many ways, and to verbally explain their thoughts to peers, allows all students to expand their repertoire and experiment with alternate models. “The more ways that children are given to think about and test out an emerging idea, the better chance it has of being integrated into a rich web of ideas and relational understanding” (Van de Walle, 2001, p. 34). A Balanced Approach: Acquiring Conceptual Understanding, Basic Skills, Math Facts, and Mental Math Strategies A conceptual understanding of mathematics allows students to develop a deep understanding of math concepts, which they can apply to a variety of real-world problems. Marian Small cites research by Carpenter and Lehrer (1999) that 5

explains conceptual understanding as “the development of understanding not only as the linking of new ideas to existing ones, but as the development of richer and more integrative knowledge structures” (Small, 2017, p. 3). A balanced math program includes the implementation of a variety of high-impact instructional practices. Math Place includes a variety of these practices throughout the lessons, including learning goals, success criteria, and descriptive feedback; direct instruction; problem-solving tasks and experiences; the use of a variety of tools and representations; and small-group instruction using flexible groupings. Math conversations are critical. Throughout the lessons, there are several Math Talks that give students opportunities to reason and prove their ideas, and listen to and debate the ideas of others. Students see math from other perspectives at the same time as they build their own mathematical understandings and feelings of confidence. It is also important for students to develop basic skills and proficiency within the different strands; for example, becoming proficient with skip-counting as they determine the total of tally marks in data. Students also gain fluency in using various mental math strategies in order to add and subtract numbers to 20. Deliberate practice plays a key role in students internalizing the skills and being able to apply them independently in new situations. By using concrete materials and discussing their ideas during Math Talks, students develop mental math strategies that help them visualize the concepts and gain automaticity of number facts and calculations. Marian Small also suggests using “rich tasks embedded in real-life experiences of children, and with rich discourse about mathematical ideas” (Small, 2017, p. 3), which aligns with Indigenous teaching that emphasizes “experiential learning, modeling, collaborative activity and teaching for meaning” (Beatty & Blair, 2015, p. 5). daltMehurseerasitn5bohg–neT1g,tah5ionlekrnfsrwdienamhegyea.monyreibnveeuentrduetssohefedraeat Math Talks There are numerous Math Talks linked to the lessons in Patterns & Relations/ Data & Probability which support the understanding of math concepts through purposeful discussion, help to reinforce and extend the learning, and offer opportunities for further investigation. (For more on Math Talks, see the Overview Guide.) In order to maximize students’ participation and active listening, you can strategically integrate the following ‘math talk moves’ into all discussions. (Adapted from Chapin, O’Connor, & Canavan Anderson, 2009) 6 Patterns & Relations/Data & Probability

Math Talk Moves Chart Example Talk Move Description Wait Time Teacher waits after posing a question before – Wait at least 10 seconds after posing a calling on a student so all students can think. question. – If a student has trouble expressing, say “Take your time.” Repeating Teacher asks students to repeat or restate what “Who can say what said in their own another student has said so more people hear words?” the idea. It encourages active listening. Revoicing Teacher restates a student’s idea to clarify and “So you are saying…. Is that what you were emphasize and then asks if the restatement is saying?” correct. This can be especially helpful for ELLs. Adding On Teacher encourages students to expand upon a “Can someone add on to what proposed idea. It encourages students to listen just said?” to peers. Reasoning Teacher asks students to respond to other “Who agrees? Who disagrees?” students’ comments by contributing and “You agree/disagree because justifying their own ideas. (sentence starter) .” aabeSaltpnBneenitmrytdusucoeosdiosemltakodetuocartnipnruhnhywasttgrgassg,woogbewGrucnillktdeeaghrehosnhteetmh.wmoondbruoteoitehtsamtvuotekutMgaehbershknieeetnerthdi.dsse,skareesnt,ysis Building Habits of Mind, Growth Mindsets, and Positive Attitudes Toward Math Math Place offers many opportunities to build and reinforce habits of mind, growth mindsets, and positive attitudes toward math, beginning with an introductory lesson that sets the tone for nurturing and developing these important skills and attitudes. (The introductory lesson, “Instilling a Growth Mindset,” can be found in the Overview Guide.) This lesson can be used at the beginning of the year to establish what the skills are and develop criteria for building them. The pertinent messages can be regularly reinforced throughout the year using the prompts and suggestions embedded in many of the Patterns & Relations/Data & Probability lessons. For interview prompts and questions to build growth mindsets and positive attitudes, see the Overview Guide or the Teacher’s Website. Embedding First Peoples Perspectives As you plan and adapt the lessons in this resource, consider how you can integrate First Peoples knowledge, stories, perspectives, and worldviews into the context of the lessons. Finding math stories outside in nature, using natural materials gathered outdoors for concrete materials, and reading stories that involve local or place-based animals and plants, help students with Indigenous ancestry see their culture reflected in their school life and classroom. 7

First Peoples Curricular Competencies Principles of Learning • U nderstanding and solving: Engage in problem-solving experiences that are connected to place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures • C onnecting and reflecting: Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts • Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place) • L earning recognizes the role of indigenous knowledge • L earning is embedded in memory, history, and story • L earning requires exploration of one’s identity When students are conceptually learning Big Ideas about patterns, data, and probability, working within a meaningful context is critical so they can connect their own experiences to the math and see its relevance. Throughout this resource, there are several opportunities to deepen understanding through other cultural lenses, including First Peoples perspectives. This approach reflects themes identified as Characteristics of Aboriginal Worldviews and Perspectives. These include: • Experiential Learning: “Look for ways to incorporate hands-on learning experiences for students” (British Columbia Ministry of Education, 2015, p. 36). Find ways to incorporate natural objects from the outdoors as students create patterns using concrete materials. • The Power of Story: Think of place-based stories that can represent the math and celebrate the cultural identities of all students. “Metaphor, analogy, example, allusion, humour, surprise, formulaic phrasing, etc., are storytelling devices that can be applied when explaining almost any non-fiction concept. Make an effort to use devices of this sort in all subject areas and to draw upon stories of the local Aboriginal community” (British Columbia Ministry of Education, 2015, p. 30). • Emphasis on Identity: “Embrace learner-centred teaching practice” (British Columbia Ministry of Education, 2015, p. 26). • Connectedness and Relationships: “Look for ways to relate learning to students’ selves, to their families and communities, and to the other aspects of Aboriginal Worldviews and Perspectives” (British Columbia Ministry of Education, 2015, p. 16). Learning is a social process, not only in the classroom but within the family and community as well. • Local Focus: Look at how Indigenous people in this area would use patterns, data they collect, and probability—in what context, for what purpose (focus on local Indigenous history, experience, stories, imagery, ecology) (British Columbia Ministry of Education, 2015, p. 22). 8 Patterns & Relations/Data & Probability

• Engagement with the Land, Nature, and the Outdoors: “Look for opportunities to get students interested and engaged with the natural world immediately available (place-based education in the area near your school). Illustrations using locally observable examples and phenomena, physical education activities, homework assignments, and student projects are examples of opportunities to promote this type of engagement” (British Columbia Ministry of Education, 2015, p. 24). In consultation with community members, think of place-based stories that can represent the math and celebrate the cultural identities of all students. Include activities that allow students to actively experience the learning. Find ways to incorporate natural objects from the outdoors as students create patterns using concrete materials. This inclusive approach allows all students to make connections between mathematics and their identities. 9